Question

The medical community unanimously agrees on the health benefits of regular exercise, but are adults listening? During each of the past 15 years, a polling organization has surveyed americans about their exercise habits. In the most recent of these polls, slightly over half of all American adults reported that they exercise for 30 or more minutes at least three times per week. The following data show the percentages of adults who reported that they exercise for 30 or more minutes at least three times per week during each of the 15 years of this study.

Year	Percentage of Adults Who Exercise 30 or more minutes at least three times per week
1	41.8
2	45.4
3	47.2
4	45.7
5	46.8
6	44.8
7	47.8
8	50.5
9	48.3
10	49.3
11	49.5
12	52.5
13	50.7
14	55
15	52.3

1. use simple linear regression to find the parameters for the line that minimizes MSE for this time series. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)
   
   y-intercept, b₀ = ______ (my answer is 46.2420)
   
   Slope, b₁ = ______ (my answer is 0.4529)
   
   MSE = ______
   
2. Use the trend equation from part (b) to forecast the percentage of adults next year (year 16 of the study) who will report that they exercise for 30 or more minutes at least three times per week. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)
   
   _____%
   
3. Use the trend equation from part (b) to forecast the percentage of adults three years from now (year 18 of the study) who will report that they exercise for 30 or more minutes at least three times per week. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)
   
   _____ %

Answer 1

The codes are given below

> t=seq(1,15,by=1)

>t
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
> x=c(41.8,45.4,47.2,45.7,46.8,44.8,47.8,50.5,48.3,49.3,49.5,52.5,50.7,55,52.3)
> x
[1] 41.8 45.4 47.2 45.7 46.8 44.8 47.8 50.5 48.3 49.3 49.5 52.5 50.7 55.0 52.3
> data=data.frame(x,t)
> fit=lm(x~t,data = data)
> fit

Call:
lm(formula = x ~ t, data = data)

Coefficients:
(Intercept) t
42.9810 0.6907

> fitted.values(fit)
1 2 3 4 5 6 7 8 9 10
43.67167 44.36238 45.05310 45.74381 46.43452 47.12524 47.81595 48.50667 49.19738 49.88810
11 12 13 14 15
50.57881 51.26952 51.96024 52.65095 53.34167
> y=fitted.values(fit)
> mse=((x-y)^2)/15
> mse
1 2 3 4 5 6 7
2.335424e-01 7.177689e-02 3.072800e-01 1.279516e-04 8.904856e-03 3.604488e-01 1.696523e-05
8 9 10 11 12 13 14
2.648919e-01 5.368617e-02 2.305707e-02 7.758867e-02 1.009381e-01 1.058800e-01 3.678683e-01
15
7.233796e-02
> mse=(sum((x-y)^2))/15
> mse
[1] 2.048346
> coef(fit)
(Intercept) t
42.9809524 0.6907143
> coef(fit)[1]
(Intercept)
42.98095
> y16=coef(fit)[1]+coef(fit)[2]*16
> y16
(Intercept)
54.03238
> y18=coef(fit)[1]+coef(fit)[2]*18
> y18
(Intercept)
55.41381

RESULTS

a)b0=42.9810

slope(b1)=0.6907

MSE=2.0483

b) The percentage of adults next year (year 16 of the study) who will report that they exercise for 30 or more minutes at least three times per week is 54.0324 %

c) The percentage of adults next year (year 18 of the study) who will report that they exercise for 30 or more minutes at least three times per week is 55.4138 %